This paper applies Thompson sampling [1]} to provide the first solution for entropic risk MAB problems which have not been previously considered before to the best of our knowledge. We proposed a new algorithm ERTS to solve this problem and proved that this proposed algorithm is asymptotically optimal for the ER MAB problem. Further work includes analysing Thompson sampling of Gaussian MABs under general risk measures and exploring Thompson sampling's performance for Entropic-Risk sub-Gaussian bandits. We may also potentially design a general framework for proving the efficacy of Thompson sampling over the state-of-the-art L/UCB-based techniques for generalised risk-averse MABs and a wider class of bandits (under reasonable assumptions, such as the crucial properties of the risk-measures, existence of conjugate prior estimates, as well as relevant concentration bounds).
